Make R the subject of the relation V = πl(R$^2$ - r$^2$)

Analyze the question: We need to rearrange V = πl(R² − r²) so that R stands alone on one side.

Concept: Use inverse operations to undo what has been done to R, working from the outside in: undo the multiplication, then the subtraction, then the square.

Solution:

1. Start with the formula: V = πl(R² − r²)

2. Divide both sides by πl: V/(πl) = R² − r²

3. Add r² to both sides: R² = V/(πl) + r²

4. Take the square root of both sides (R is a positive length): R = √(V/(πl) + r²)

Checking the options:

• A: √(V/(πl) + r²) matches our result.
• B has a minus sign, which is wrong because r² is added, not subtracted.
• C and D fail to divide V by πl properly. They multiply πlr² instead.

Answer: A

Tip: When making a variable the subject, "undo" operations in reverse order of BODMAS. Here the bracket was multiplied by πl, so divide first. Then move −r² across, which becomes +r². Finally, take the square root. Always remember that moving a term across the equals sign changes its sign.